Equation For Combined Time To Build A Patio
Introduction
In the realm of mathematics, problems often present themselves as puzzles, challenging us to decipher the relationships between different quantities and arrive at a solution. One such problem involves the collaborative effort of two individuals, Franklin and Scott, in constructing a cement patio. Franklin, a seasoned craftsman, takes 14 hours to complete a 200-square-foot patio, while Scott, with his own set of skills, accomplishes the same task in 10 hours. The question at hand is to determine the equation that will help us find x, the number of hours it would take for Franklin and Scott to construct the patio if they were to work together. This problem delves into the concept of combined work rates and how they contribute to the overall time required to complete a task. To solve this, we need to understand how each person's individual work rate contributes to their combined work rate. The problem is not just a mathematical exercise; it's a reflection of real-world scenarios where teamwork and collaboration can significantly impact the efficiency and speed of project completion. This introduction sets the stage for a deeper exploration of the problem, guiding us through the process of identifying the key variables, understanding their relationships, and ultimately formulating the equation that will unlock the solution. By understanding the underlying principles of work rates and their combined effect, we can not only solve this specific problem but also apply the same logic to a variety of similar situations, making it a valuable skill in both academic and practical contexts.
Understanding Individual Work Rates
Before we can determine the equation to find the combined time it takes Franklin and Scott to build the patio, we must first understand their individual work rates. The work rate is a measure of how much of a task an individual can complete in a given unit of time. In this scenario, the task is building a 200-square-foot patio, and the unit of time is hours. To calculate Franklin's work rate, we divide the amount of work completed (1 patio) by the time it takes him (14 hours). This gives us a work rate of 1/14 of the patio per hour. Similarly, Scott's work rate is calculated by dividing the amount of work completed (1 patio) by the time it takes him (10 hours), resulting in a work rate of 1/10 of the patio per hour. These individual work rates are crucial because they form the foundation for understanding how much work each person contributes when they work together. It's essential to recognize that work rate is an inverse relationship with time; the faster someone works (i.e., the less time they take), the higher their work rate. Understanding this concept is vital for setting up the correct equation to solve the problem. When individuals work together, their work rates combine to determine the overall rate at which the task is completed. By calculating each person's individual contribution, we can then add these contributions together to find their combined work rate, which will ultimately help us determine the time it takes for them to complete the patio together. This step-by-step approach, starting with individual work rates, is key to successfully navigating this type of problem and arriving at an accurate solution.
Combining Work Rates for Collaborative Efforts
Now that we know Franklin's work rate is 1/14 of the patio per hour and Scott's work rate is 1/10 of the patio per hour, the next step is to combine these rates to find their combined work rate. When Franklin and Scott work together, their individual efforts contribute to the overall progress of the patio construction. To find their combined work rate, we simply add their individual work rates together. This means we add 1/14 (Franklin's work rate) and 1/10 (Scott's work rate). The sum represents the fraction of the patio they can complete together in one hour. Adding fractions requires a common denominator, so we need to find the least common multiple (LCM) of 14 and 10, which is 70. We then convert both fractions to have this common denominator: (1/14) * (5/5) = 5/70 and (1/10) * (7/7) = 7/70. Adding these together gives us 5/70 + 7/70 = 12/70, which can be simplified to 6/35. This means that together, Franklin and Scott can complete 6/35 of the patio in one hour. This combined work rate is a crucial piece of information because it directly relates to the time it takes for them to complete the entire patio together. The combined work rate represents the efficiency of their collaboration, taking into account each person's individual contribution. Understanding how to combine work rates is essential not only for this particular problem but also for various real-world scenarios where multiple individuals collaborate on a task. By accurately calculating the combined work rate, we can predict the time it will take to complete the task, allowing for better planning and resource allocation. This concept is applicable in various fields, from construction and manufacturing to project management and software development. Therefore, mastering the ability to combine work rates is a valuable skill for both academic and practical applications.
Formulating the Equation to Determine Combined Time
With the combined work rate of Franklin and Scott established at 6/35 of the patio per hour, we can now formulate the equation to find x, the number of hours it will take them to complete the patio together. The fundamental principle here is that work rate multiplied by time equals the amount of work done. In this case, the amount of work done is the completion of one whole patio. Therefore, we can set up the equation as follows: (Combined Work Rate) * (Time) = 1. Substituting the combined work rate we calculated earlier, we get (6/35) * x = 1. This equation represents the mathematical relationship between their combined work rate, the time they work together, and the completion of the entire task. To solve for x, we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by the reciprocal of 6/35, which is 35/6. This gives us: x = 1 * (35/6). Simplifying this, we find that x = 35/6 hours. This fraction can be converted to a mixed number or a decimal to provide a more practical understanding of the time required. 35/6 is equal to 5 and 5/6 hours, or approximately 5.83 hours. Therefore, the equation (6/35) * x = 1 is the key to unlocking the solution to our problem. It encapsulates the combined efforts of Franklin and Scott and allows us to determine the time it takes for them to build the patio together. This equation is a powerful tool that demonstrates how mathematical principles can be applied to real-world situations, providing a clear and concise way to model and solve problems involving collaborative work.
Solving for x and Interpreting the Result
Having formulated the equation (6/35) * x = 1, we can now solve for x to determine the number of hours it would take Franklin and Scott to build the patio together. As previously mentioned, to isolate x, we multiply both sides of the equation by the reciprocal of 6/35, which is 35/6. This gives us: x = 1 * (35/6), which simplifies to x = 35/6 hours. To interpret this result in a more practical way, we can convert the improper fraction 35/6 into a mixed number. Dividing 35 by 6, we get 5 with a remainder of 5, so 35/6 is equal to 5 and 5/6 hours. This means that it would take Franklin and Scott 5 full hours and 5/6 of an hour to complete the patio together. To further refine our understanding, we can convert the fractional part of an hour into minutes. Since there are 60 minutes in an hour, 5/6 of an hour is (5/6) * 60 = 50 minutes. Therefore, it would take Franklin and Scott 5 hours and 50 minutes to build the patio together. This result highlights the efficiency gained through collaboration. By working together, Franklin and Scott can complete the patio in significantly less time than either of them could individually. This is because their combined work rate is higher than either of their individual work rates. The solution also demonstrates the practical application of mathematical concepts in real-world scenarios. By understanding the relationship between work rate, time, and the amount of work done, we can solve problems involving collaborative efforts and make informed decisions about resource allocation and project planning. This ability to interpret mathematical results in a meaningful context is a valuable skill that can be applied in various aspects of life and work.
Conclusion
In conclusion, the problem of determining the time it would take Franklin and Scott to build a 200-square-foot cement patio together highlights the power of mathematical problem-solving in real-world scenarios. By understanding the concepts of individual work rates, combined work rates, and their relationship to time, we were able to formulate and solve an equation that accurately predicted the outcome of their collaboration. The key to solving this problem was breaking it down into manageable steps: first, calculating the individual work rates of Franklin and Scott; then, combining these rates to find their combined work rate; and finally, using this combined work rate to formulate an equation that related time to the completion of the task. The resulting equation, (6/35) * x = 1, allowed us to solve for x, which represented the number of hours it would take Franklin and Scott to build the patio together. Our solution, 5 hours and 50 minutes, demonstrated the efficiency gained through teamwork and collaboration. This problem serves as a valuable example of how mathematical principles can be applied to practical situations, providing insights into resource allocation, project planning, and the benefits of collaborative efforts. Furthermore, the process of solving this problem reinforces the importance of critical thinking, logical reasoning, and the ability to translate real-world scenarios into mathematical models. These skills are essential not only for academic success but also for navigating the complexities of everyday life and making informed decisions in various professional settings. By mastering the concepts and techniques demonstrated in this problem, individuals can enhance their problem-solving abilities and approach challenges with confidence and clarity.
